Problem 363 - Precalculus

Question

Find the length of a circular arc with a radius 12 centimeters subtended by the central angle of \(30^{\circ} .\)

Step-by-Step Solution

Answer

The arc length is \(2\pi\) cm.

Step 1: Understand the Problem

We need to find the length of an arc of a circle. The given circle has a radius of 12 cm and the central angle subtending the arc is \(30^{\circ}\).

Step 2: Convert Angle to Radians

To use the arc length formula, we first convert the angle from degrees to radians since the formula uses radians. Recall: \( 1^{\circ} = \frac{\pi}{180} \) radians. Thus, \(30^{\circ} = 30 \times \frac{\pi}{180} = \frac{\pi}{6}\) radians.

Step 3: Apply Arc Length Formula

The formula for the arc length \(L\) of a circle is \(L = r \theta\), where \(r\) is the radius and \(\theta\) is the angle in radians. Here, \(r = 12\) cm, and \(\theta = \frac{\pi}{6}\) radians.

Step 4: Calculate the Arc Length

Substitute the values into the formula: \( L = 12 \times \frac{\pi}{6} = 2\pi \) cm.

Key Concepts

Radians

Radians are a method to measure angles based on the radius of a circle. Unlike degrees, where a full circle is divided into 360 parts, radians divide a circle into a measure that relates directly to the circle's properties.

• One radian is the angle created when the arc length equals the radius of the circle.
• The formula to convert degrees to radians is crucial: \( 1^{\circ} = \frac{\pi}{180} \) radians.
• For instance, in the exercise, the central angle of \(30^{\circ}\) converts to \(\frac{\pi}{6}\) radians.

Understanding this conversion is crucial because many mathematical formulas, including those related to circular motion and trigonometry, are most naturally expressed in radians rather than degrees.
With this knowledge, solving problems related to circles becomes more intuitive.

Central Angle

The central angle is an angle whose vertex is at the center of a circle. This angle divides the circle into two arcs: a major arc and a minor arc.

• The central angle is key to finding the length of an arc.
• In our step-by-step solution, knowing the central angle in radians allows us to use the arc length formula: \( L = r \theta \).
• It is significant in geometry because it determines the proportionality of the arc length to the radius.

In practical problems, once you know the central angle and the radius, you can determine any other circular property involving arcs, making the central angle a fundamental aspect of circle geometry.

Circle

Circles appear everywhere in mathematics, with critical properties like centers, radii, diameters, and arcs. Each of these properties aids in solving different problems.

• The circle is defined by its central point and all points equidistant, known as the radius.
• The perimeter of a circle is known as the circumference, which is \( 2\pi r \).
• Arcs are segments of the circle's circumference, characterized by its length, dependent on the central angle.

Understanding the fundamental properties of a circle is essential for solving geometry problems.
For example, the arc length is a specific part of the circumference, and as demonstrated in the solution, knowing the radius and central angle, we can find it easily. Circles and their properties form the basis for much of geometry and trigonometry.